Mix kimchi and shredded chicken together. This way, any slight kimchi juice will be absorbed by the chicken.
Add butter to a large skillet over medium-high heat. Place tortilla open face on the pan. On one side of the tortilla; add kimchi, chicken, cheese, sesame oil, and salt & pepper (to taste). Fold tortilla in half and cook until one side starts to turn golden brown (about 2 minutes per side). Once browned on one side, flip to the other side to brown.
Slice into wedges and serve with a dollop of sour cream!